From the observation deck of the lighthouse at Sasquatch Point 70 feet above the surface of Lake Ippizuti, a lifeguard spots a boat out on the lake sailing directly toward the lighthouse. The first sighting had an angle of depression of 8.3° and the second sighting had an angle of depression of 25.6°. How far had the boat traveled between the sightings? Round your answer to the nearest foot.
 



To solve the problem, we'll use trigonometry. The angles of depression can help us determine the horizontal distances from the lighthouse to the boat during the two sightings.

### Step 1: Determine the horizontal distances during both sightings.

Let's define:
- \( h = 70 \) feet (height of the lighthouse above the lake).
- \( d_1 \) = horizontal distance from the lighthouse to the boat at the first sighting.
- \( d_2 \) = horizontal distance from the lighthouse to the boat at the second sighting.
- \( \theta_1 = 8.3^\circ \) (angle of depression for the first sighting).
- \( \theta_2 = 25.6^\circ \) (angle of depression for the second sighting).

Using the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

For the first sighting:
\[
\tan(8.3^\circ) = \frac{70}{d_1}
\]
\[
d_1 = \frac{70}{\tan(8.3^\circ)}
\]

For the second sighting:
\[
\tan(25.6^\circ) = \frac{70}{d_2}
\]
\[
d_2 = \frac{70}{\tan(25.6^\circ)}
\]

### Step 2: Calculate the values.

We'll calculate \( d_1 \) and \( d_2 \) and then find the distance the boat traveled between the two sightings.

\[
d_1 = \frac{70}{\tan(8.3^\circ)} \quad \text{and} \quad d_2 = \frac{70}{\tan(25.6^\circ)}
\]

Let's compute these distances.The boat traveled approximately 334 feet between the two sightings.

Would you like more details or have any further questions? Here are some related questions you might find interesting:

1. How can the angle of depression be used to calculate distances in other scenarios?
2. What other trigonometric functions could be useful in similar problems?
3. How does the height of the lighthouse affect the distance calculation?
4. What happens to the horizontal distance if the angle of depression is very small?
5. How would this problem change if the observer were at a different height?

**Tip:** When solving problems involving angles of elevation or depression, always ensure you correctly interpret the angle as measured from the horizontal, not from the vertical.

Learn how to calculate the distance a boat travels using angles of depression in this math problem solution. Given observations from a lighthouse, determine the horizontal distance between sightings using trigonometry. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation